E7 polytope

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Orthographic projections in the E7 Coxeter plane
Up2 3 21 t0 E7.svg
321
CDel nodea 1.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
Up2 2 31 t0 E7.svg
231
CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea 1.png
Up2 1 32 t0 E7.svg
132
CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 01lr.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png

In 7-dimensional geometry, there are 127 uniform polytopes with E7 symmetry. The three simplest forms are the 321, 231, and 132 polytopes, composed of 56, 126, and 576 vertices respectively.

Geometry branch of mathematics that measures the shape, size and position of objects

Geometry is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space. A mathematician who works in the field of geometry is called a geometer.

Uniform 7-polytope vertex-transitive 7-polytope bounded by uniform facets

In seven-dimensional geometry, a 7-polytope is a polytope contained by 6-polytope facets. Each 5-polytope ridge being shared by exactly two 6-polytope facets.

3<sub> 21</sub> polytope uniform 7-polytope

In 7-dimensional geometry, the 321 polytope is a uniform 7-polytope, constructed within the symmetry of the E7 group. It was discovered by Thorold Gosset, published in his 1900 paper. He called it an 7-ic semi-regular figure.

They can be visualized as symmetric orthographic projections in Coxeter planes of the E7 Coxeter group, and other subgroups.

Orthographic projection form of parallel projection in which all the projection lines are orthogonal to the projection plane

Orthographic projection is a means of representing three-dimensional objects in two dimensions. It is a form of parallel projection, in which all the projection lines are orthogonal to the projection plane, resulting in every plane of the scene appearing in affine transformation on the viewing surface. The obverse of an orthographic projection is an oblique projection, which is a parallel projection in which the projection lines are not orthogonal to the projection plane.

Graphs

Symmetric orthographic projections of these 127 polytopes can be made in the E7, E6, D6, D5, D4, D3, A6, A5, A4, A3, A2 Coxeter planes. Ak has k+1 symmetry, Dk has 2(k-1) symmetry, and E6 and E7 have 12, 18 symmetry respectively.

For 10 of 127 polytopes (7 single rings, and 3 truncations), they are shown in these 9 symmetry planes, with vertices and edges drawn, and vertices colored by the number of overlapping vertices in each projective position.

# Coxeter plane graphs Coxeter diagram
Schläfli symbol
Names
E7
[18]
E6A6
[7x2]
A5
[6]
A4 / D6
[10]
D5
[8]
A2 / D4
[6]
A3 / D3
[4]
1 Up2 2 31 t0 E7.svg Up2 2 31 t0 E6.svg Up2 2 31 t0 A6.svg Up2 2 31 t0 A5.svg Up2 2 31 t0 D6.svg Up2 2 31 t0 D5.svg Up2 2 31 t0 D4.svg Up2 2 31 t0 D3.svg CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea 1.png
231 (laq)
2 Up2 2 31 t1 E7.svg Up2 2 31 t1 E6.svg Up2 2 31 t1 A6.svg Up2 2 31 t1 A5.svg Up2 2 31 t1 D6.svg Up2 2 31 t1 D5.svg Up2 2 31 t1 D4.svg Up2 2 31 t1 D3.svg CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea 1.pngCDel 3a.pngCDel nodea.png
Rectified 231 (rolaq)
3 Up2 1 32 t1 E7.svg Up2 1 32 t1 E6.svg Up2 1 32 t1 A6.svg Up2 1 32 t1 A5.svg Up2 1 32 t1 D6.svg Up2 1 32 t1 D5.svg Up2 1 32 t1 D4.svg Up2 1 32 t1 D3.svg CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 10.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
Rectified 132 (rolin)
4 Up2 1 32 t0 E7.svg Up2 1 32 t0 E6.svg Up2 1 32 t0 A6.svg Up2 1 32 t0 A5.svg Up2 1 32 t0 D6.svg Up2 1 32 t0 D5.svg Up2 1 32 t0 D4.svg Up2 1 32 t0 D3.svg CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 01lr.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
132 (lin)
5 Up2 3 21 t2 E7.svg Up2 3 21 t2 E6.svg Up2 3 21 t2 A6.svg Up2 3 21 t2 A5.svg Up2 3 21 t2 D6.svg Up2 3 21 t2 D5.svg Up2 3 21 t2 D4.svg Up2 3 21 t2 D3.svg CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea 1.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
Birectified 321 (branq)
6 Up2 3 21 t1 E7.svg Up2 3 21 t1 E6.svg Up2 3 21 t1 A6.svg Up2 3 21 t1 A5.svg Up2 3 21 t1 D6.svg Up2 3 21 t1 D5.svg Up2 3 21 t1 D4.svg Up2 3 21 t1 D3.svg CDel nodea.pngCDel 3a.pngCDel nodea 1.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
Rectified 321 (ranq)
7 Up2 3 21 t0 E7.svg Up2 3 21 t0 E6.svg Up2 3 21 t0 A6.svg Up2 3 21 t0 A5.svg Up2 3 21 t0 D6.svg Up2 3 21 t0 D5.svg Up2 3 21 t0 D4.svg Up2 3 21 t0 D3.svg CDel nodea 1.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
321 (naq)
8 Up2 2 31 t01 E7.svg Up2 2 31 t01 E6.svg Up2 2 31 t01 A6.svg Up2 2 31 t01 A5.svg Up2 2 31 t01 D6.svg Up2 2 31 t01 D5.svg Up2 2 31 t01 D4.svg Up2 2 31 t01 D3.svg CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea 1.pngCDel 3a.pngCDel nodea 1.png
Truncated 231 (talq)
9 Up2 1 32 t01 E7.svg Up2 1 32 t01 E6.svg Up2 1 32 t01 A6.svg Up2 1 32 t01 A5.svg Up2 1 32 t01 D6.svg Up2 1 32 t01 D5.svg Up2 1 32 t01 D4.svg Up2 1 32 t01 D3.svg CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 11.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
Truncated 132 (tilin)
10 Up2 3 21 t01 E7.svg Up2 3 21 t01 E6.svg Up2 3 21 t01 A6.svg Up2 3 21 t01 A5.svg Up2 3 21 t01 D6.svg Up2 3 21 t01 D5.svg Up2 3 21 t01 D4.svg Up2 3 21 t01 D3.svg CDel nodea 1.pngCDel 3a.pngCDel nodea 1.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
Truncated 321 (tanq)

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References

Harold Scott MacDonald Coxeter Canadian mathematician

Harold Scott MacDonald "Donald" Coxeter, FRS, FRSC, was a British-born Canadian geometer. Coxeter is regarded as one of the greatest geometers of the 20th century. He was born in London, received his BA (1929) and PhD (1931) from Cambridge, but lived in Canada from age 29. He was always called Donald, from his third name MacDonald. He was most noted for his work on regular polytopes and higher-dimensional geometries. He was a champion of the classical approach to geometry, in a period when the tendency was to approach geometry more and more via algebra.

International Standard Book Number Unique numeric book identifier

The International Standard Book Number (ISBN) is a numeric commercial book identifier which is intended to be unique. Publishers purchase ISBNs from an affiliate of the International ISBN Agency.

Fundamental convex regular and uniform polytopes in dimensions 2–10
Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron OctahedronCube Demicube DodecahedronIcosahedron
Uniform 4-polytope 5-cell 16-cellTesseract Demitesseract 24-cell 120-cell600-cell
Uniform 5-polytope 5-simplex 5-orthoplex5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex6-cube 6-demicube 122221
Uniform 7-polytope 7-simplex 7-orthoplex7-cube 7-demicube 132231321
Uniform 8-polytope 8-simplex 8-orthoplex8-cube 8-demicube 142241421
Uniform 9-polytope 9-simplex 9-orthoplex9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplexn-cube n-demicube 1k22k1k21 n-pentagonal polytope
Topics: Polytope familiesRegular polytopeList of regular polytopes and compounds